Quantum supersymmetric pairs and the Serre relations via $\mathrm i$Hopf algebras

Abstract

We study iHopf algebras associated with quantum supergroups of basic type. For a quantum supersymmetric pair $(\mathbf{U}, \mathbf{U}^\imath)$, we realize $\mathbf{U}^\imath$ and $\mathbf{U}$ as the iHopf algebras of the Borel quantum group ${\hat{\mathbf{U}}}_q^{\geq0}$ and the tensor product algebra ${\hat{\mathbf{U}}}_q^{\geq0}\otimes {\hat{\mathbf{U}}}_q^{\geq0}$, respectively, while the coideal subalgebra structure is encoded by an embedding between the iHopf algebras. Moreover, we derive an explicit conversion formula between the iHopf multiplication and the original multiplication, which provides a general mechanism transforming the defining relations of quantum (super)groups into relations of iquantum (super)groups. In particular, all the Serre relations that occur in quasi-split iquantum supergroups of basic type are characterized.